An automated subtraction of NLO EW infrared divergences
Abstract
In this paper a generalisation of the Catani–Seymour dipole subtraction method to nexttoleading order electroweak calculations is presented. All singularities due to photon and gluon radiation off both massless and massive partons in the presence of both massless and massive spectators are accounted for. Particular attention is paid to the simultaneous subtraction of singularities of both QCD and electroweak origin which are present in the nexttoleading order corrections to processes with more than one perturbative order contributing at Born level. Similarly, embedding nondipolelike photon splittings in the dipole subtraction scheme discussed. The implementation of the formulated subtraction scheme in the framework of the Sherpa MonteCarlo event generator, including the restriction of the dipole phase space through the \(\alpha \)parameters and expanding its existing subtraction for NLO QCD calculations, is detailed and numerous internal consistency checks validating the obtained results are presented.
1 Introduction
As RunI of the LHC has been successfully completed, culminating in the celebrated experimental confirmation of the existence of the Higgs boson, RunII proceeds its datataking at the unprecedented centreofmass energy of 13 TeV. As the much anticipated discovery of signals of beyondtheStandardModel physics is still lacking, precision tests scrutinising the Standard Model are of prime importance, now and in the foreseeable future. At the same time, new physics searches are looking for increasingly small signals demanding more precise estimates of the Standard Model backgrounds. This expansion of sensitivity of both precision measurements and new physics searches in the multiTeV region demand an immense improvement of theoretical predictions.
This precision can be achieved by the inclusion of nextto and nexttonexttoleading order (NLO and NNLO) corrections in the strong coupling and nexttoleading order electroweak (EW) corrections. Here it should be noted that both NNLO QCD and NLO EW corrections are expected to be of a similar magnitude for inclusive observables as numerically \(\alpha _s^2\approx \alpha \). On selected differential distributions, however, electroweak corrections can grow much larger. They are dominated by photon emissions in the distributions of final state leptons, for example. In invariant mass spectra of lepton pairs below a resonance, for example, O(1) corrections can be present, in which case a proper resummation should be included [1]. Similarly, looking at the (multi)TeV regime, the NLO EW corrections quickly grow considerably, reducing cross sections by a few tens of percent, due to the emergence of large electroweak Sudakov corrections arising as the scattering energies \(Q^2\gg m_W^2\) [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. In this regime they are larger than even the NLO QCD corrections in many cases and their omission becomes the dominant uncertainty in experimental studies and searches.
To this end, benefiting from the wellestablished techniques developed for the automation of NLO QCD corrections many NLO EW corrections have been calculated recently [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38]. To fully automate these computations at NLO EW accuracy in a MonteCarlo framework all infrared divergences need to be regulated, where various incarnations of subtraction methods have proven to be the methods of choice for practical implementations [39, 40, 41, 42, 43]. Similar subtractions have also been published for NLO EW calculations using a dipole picture [1, 44, 45, 46]. Contrary to the QCD case, only the implementation of [46], restricted to photon emissions of fermions, is publicly available though.
Besides the generalisation to all divergent splittings at \(\mathcal {O}(\alpha )\), including photon splittings and photon emissions off massive scalars and vector bosons, this publication addresses the issue of automatically detecting simultaneously occurring QCD and QED singularities and subtracting them consistently. These occur as soon as the Born process is defined at multiple orders \(\mathcal {O}(\alpha _s^n\alpha ^{Nn})\). In this case the NLO EW correction to the \(\mathcal {O}(\alpha _s^n\alpha ^{Nn})\) process, being of \(\mathcal {O}(\alpha _s^n\alpha ^{Nn+1})\), is at the same time the NLO QCD correction to the \(\mathcal {O}(\alpha _s^{n1}\alpha ^{Nn+1})\) process and will in general exhibit the corresponding infrared singularities. Further, matters of the organisation of the contributing partonic processes and their mapping to reduce the computational complexity along with the provision of infrared safe phase space cuts are discussed. The algorithm is implemented in the Amegic [47] matrix element generator which is part of the Sherpa [48] MonteCarlo event generator framework. It bases on the automated subtraction of massless NLO QCD divergences therein [49, 50]. The implementation presented in this publication has, in various preliminary forms, already been used to calculate electroweak corrections to a multitude of important signal and background processes [19, 21, 25, 31, 32, 34, 37, 51, 52, 53], highlighting its versatility.
2 Catani–Seymour subtraction at NLO EW
All infrared divergences that occur at NLO EW are of QED origin. No subtractions of potentially large, but finite, corrections involving the emissions of real and virtual massive electroweak gauge bosons will be considered. The practical implementation described in Sect. 3 follows the general lines of [49, 50].
2.1 Differential subtraction terms
Since the QED charges are real numbers, the chargecorrelator simply multiplies the matrix element and only leaves the spincorrelators \(\mathbf {V}_{ij,k}\), \(\mathbf {V}_{ij}^a\), \(\mathbf {V}_{j,k}^a\) and \(\mathbf {V}_j^{a,b}\) as insertions in the spincorrelated underlying Born matrix elements. The spincorrelators directly correspond to their QCD counterparts and are detailed in Appendix A. It also defines the initial state momentum rescaling parameters \(x_{ij,a}\), \(x_{aj,k}\) and \(x_{aj,b}\), as well as the splitting variables \(y_{ij,k}\), \(u_j\) and \(v_j\). The \(\{\alpha _\text {dip}\}=\{\alpha _\text { FF},\alpha _\text { FI},\alpha _\text { IF},\alpha _\text { II}\}\) parameters serve to restrict the phase space where the individual dipole terms are nonzero and therefore need to be evaluated [54, 55]. They are constructed such that for every \(\alpha _{{\widetilde{\imath \jmath }}{\tilde{k}}}>0\) the singularity is fully subtracted. The introduction of a parameter \(\kappa \) in dipoles where a final state photon splits into a massive fermion pair in the presence of a final state spectator similarly allows a redistribution of finite terms, cf. Appendix A.
2.2 Integrated subtraction terms
The divergences of the \({{\varvec{I}}}\) operator are encoded in the functions \(\mathcal {V}_{ik}\) and \(\Gamma _i\). While the former contains all soft(quasi)collinear divergences the latter contains the pure (quasi)collinear ones. They do not only differentiate whether the emitter is a photon or not, but also between different spins of the emitter. Their precise form as well as the the flavour constants \(\gamma _i\) and \(K_i\) are given in Appendix B. \(A_{ik}^I\) encodes the dependence on the phase space restriction of the individual dipoles \(\{\alpha _\text {dip}\}\). Finite terms originating in dipoles involving initial state legs, however, can be pushed into the \({{\varvec{K}}}\) operator. Thus, \(A_{ik}^I\) by convention only depends on \(\alpha _\text { FF}\). Its precise form is given in Appendix C.
The \({{\varvec{K}}}\) and \({{\varvec{P}}}\) operators. The \({{\varvec{K}}}\) and \({{\varvec{P}}}\) operators collect all pieces of the integrated dipole terms that are not collected in the \({{\varvec{I}}}\) operator and combines them with the collinear counterterms \(\mathrm {C}\) to give a finite result as \(\epsilon \rightarrow 0\). By construction they contain only remainders of splittings where either the emitter or the spectator is in the initial state. Thus, they are comprised of terms arising due to the change of the flavour or the partonic momentum fraction x of an initial state due to a splitting.
3 Implementation
The implementation of the QED generalisation of the Catani–Seymour dipole subtraction scheme in Sherpa ’s matrix element generator Amegic proceeds along the lines of [49]. As in general real and virtual corrections of \(O(\alpha _s^n\alpha ^m)\) contain divergences of both QCD and QED origin, both cases are included in this section. In the following, the general structure of the implementation is reviewed.
3.1 Identification of dipoles
The starting point to construct the involved subtraction terms in the Catani–Seymour subtraction formalism is a given flavour configuration in the Born or the real emission phase space and the perturbative order \(\mathcal {O}(\alpha _s^n\alpha ^m)\) in accordance with the respective virtual or real correction to be computed. For all parts, onthefly variations of both the factorisation scale \(\mu _\text {F}\) and the renormalisation scale \(\mu _\text {R}\) are available through an extension of the algorithm detailed in [56].
 1.
Based on the quantum numbers and flavours of the triplet it is decided whether a QCD, a QED or both splitting function can exist. A QCD splitting function can exist only if i, j and k are colour charged, while a QED splitting function can exist only if the chargecorrelator \({{\varvec{Q}}}_{{\widetilde{\imath \jmath }}{\tilde{k}}}^2\) does not vanish. The dipole type is determined based on whether i and k are in the initial or final state. A given triplet \(\{i,j,k\}\) may exhibit both QCD and QED divergences, and thus may form both a QCD and a QED dipole.
 2.
The flavours \({\widetilde{\imath \jmath }}\) and \({\tilde{k}}\) are determined for each possible splitting function.
 3.
For each possible splitting function \(\{{\widetilde{\imath \jmath }},{\tilde{k}}\}\rightarrow \{i,j,k\}\) the underlying Born configuration and its order, \(O(\alpha _s^{n1}\alpha ^m)\) in case of a QCD splitting function and \(O(\alpha _s^n\alpha ^{m1})\) in case of a QED splitting function, are determined. If, including the insertion of the appropriate colour, charge and spincorrelations, such a process at this order exists, a dipole subtraction term is built.
Integrated subtraction terms. Similar to the above discussed real emission corrections, the virtual correction configurations \(\{ab\}\rightarrow \{1,\ldots ,m\}\) at order \(O(\alpha _s^n\alpha ^m)\) in general exhibits poles due to both QCD and QED origins. To subtract them, both QCD and QED integrated subtraction need to be included. In fact, they naturally arise as counterparts to differential subtraction terms constructed for the corresponding real emission correction, as guaranteed by Bloch and Nordsieck [57] or Kinoshita–Lee–Nauenberg [58, 59] theorem. Consequently, QCD and QED \({{\varvec{I}}}, {{\varvec{K}}}\) and \({{\varvec{P}}}\) operators are constructed. While their QCD version are described in detail in [49, 50], their QED version of Eq. (2.6) are discussed below.
The thus transformed form of the \({{\varvec{K}}}\) and \({{\varvec{P}}}\) operators require only a single evaluation of the potentially costly matrix elements in \(\mathrm {B}\) while retaining the number of computations of PDFs needed, speeding up the computation considerably for involved processes. This allows to generate and evaluate the underlying Born matrix element for both the \({{\varvec{I}}}\) operators and the \({{\varvec{K}}}\) and \({{\varvec{P}}}\) operators at the same time. In fact, due to these three operators being simple multiplicative scalars, the common underlying Born matrix element is identical to the standard Born matrix element, allowing for their simultaneous calculation at no extra cost. The operators themselves are then built from dipoles constructed from all doublets \(\{i,k\}\), \(i\ne k\), of external partons available in the partonic process for which the chargecorrelator \({{\varvec{Q}}}_{ik}^2\) does not vanish.
3.2 External photons
External photons can play different roles in a calculation: they can either be resolved or unresolved. According to this distinction they should be treated differently at NLO EW [31, 60]. Initial state photons are always unresolved at a hadron collider. They thus should be treated in a shortdistance scheme, allowing them to split into massless fermions, necessitating a proper subtraction of infrared divergences. Final state photons on the other hand can play both roles. If they are considered resolved, they should be treated in an onshell scheme and no explicit photon splitting is allowed. Concerning the dipole subtraction discussed in this paper they are thus neutral particles and do not form part of a dipole, except as possible recoil partner of another unresolved photon. A final state unresolved photon, on the other hand, again must be treated in a shortdistance scheme, necessitating their splittings to be subtracted.
 0.
only allow initial state partons as spectators,
 1.
only allow final state partons as spectators,
 2.
only allow QED charged particles as spectators,
 3.
only allow QED neutral particles as spectators,
 4.
allow all particles as spectators.
3.3 Process mappings
In general, physical cross sections include multiple different partonic channels. However, many of these partonic channels share identical squared matrix elements, potentially differing by constant factors. They, thus, do not need to be recomputed for every flavour channel but can be reused. As a typical example, consider the production of a lepton pair in association with two jets. The process \(g\,d\rightarrow e^+e^\,g\,d\) shares a common squared matrix element with \(g\,s\rightarrow e^+e^\,g\,s\) and \(g\,b\rightarrow e^+e^\,g\,b\) on Born level at \(\mathcal {O}(\alpha _s^2\alpha ^2)\). Hence, the squared matrix elements of the latter two partonic processes are mapped on the first, reusing its computed value. In this way, of the 95 partonic channels of this process, only 30 have to be computed. Further mappings of individual graphs and subgraphs are implemented but are not discussed in the following, see [47, 48].
In Sherpa ’s matrix element generators various forms of process mappings are implemented to reduce the computational complexity and memory footprint, both for the virtual and real emission corrections, cf. [47, 48, 49, 50]. However, while for NLO QCD corrections to the leading order Born process it is true that if the Born process is mappable onto another existing process, then also both the virtual correction and the insertionoperatoraugmented colourcorrelated underlying Born process of the integrated subtraction term are mappable to the same process. This is no longer true when considering the NLO EW or NLO QCD corrections to subleading Born processes.

The real emission process and its associated dipole subtraction terms are grouped in one computational unit. A given partonic channel of the real emission process can be mapped onto another already existing one if both processes consist of the same diagrams and all involved (internal and external) particles have the same masses and widths and the same underlying interaction (coupling factors may differ by a constant). Is this the case the whole computational unit can be mapped and the result of the mappedto process can simply be reused.

Individual dipoles can be mapped if the emitter, emittee and spectator indices are identical, and the underlying Born process can be mapped according to the above rules. In this case, the result of the mappedto process can simply be reused.

Underlying Born matrix elements can be mapped if the Bornlevel emitter \({\widetilde{\imath \jmath }}\) carries the same indices and the underlying Born process itself can be mapped. The spin correlation insertion operator, needed if parton \({\widetilde{\imath \jmath }}\) is either a gluon or a photon and described in [49], is encoded in the calculational routines and necessitates the above restriction. Can the underlying Born process be mapped, only the calculational routines can be shared, reducing the memory footprint, but due to the potentially differing underlying Born momenta its result has to be recomputed.

The virtual correction process, interfaced from an external virtual correction provider, as well as its associated integrated dipole subtraction terms and the collinear counterterm, taken together and reformulated according to Sect. 3.1, are grouped into one computational unit. If the underlying Born processes of the integrated subtraction contribution (in all orders required) can be mapped according to the above rules and the virtual correction provider confirms that the virtual correction can be mapped onto the same process that Sherpa ’s treelevel matrix element generator maps the underlying Born processes onto, the whole computational unit is mapped. In this case, the result of the mappedto process can simply be reused. The \({{\varvec{K}}}\) and \({{\varvec{P}}}\) operators, whose internal PDF factors depend on the initial state flavour, still have to be recomputed, but their matrix element coefficients are cached.

If the virtual correction provider cannot confirm the mapping performed for the underlying Born process, only the underlying Born process is mapped and virtual correction is recomputed. Again, the \({{\varvec{K}}}\) and \({{\varvec{P}}}\) have to be recomputed, but their matrix element coefficients are cached. Here, efficiency is lost if the virtual correction provider uses less efficient process mappings.
3.4 Fiducial phase space definition
Phase space restrictions are an essential part of the implementation of a framework for automated NLO calculcations. These cuts, however, need to be applied in an infrared safe way. At NLO, they must not discriminate between a massless parton before and after its collinear splitting or before and after a soft gluon or photon emission. Thus, if QCD singularities are present, massless QCD charged particles must be clustered into jets before any further cuts are applied. Similarly, in case of the presence of QED divergences, massless charged particles must either be dressed with the surrounding photons or be included in the jet algorithm. Subtleties arise in the presence of both QCD and QED singularities simultaneously. Here, usually, only a fully democratic jet finding can consistently treat all singularities, although specialised solutions exist for simplified situations. Massive QCD and QED charged particles may be treated as bare as their mass shields the collinear singularity, but can also be included into jet finding and dressing algorithms. An intermediate scheme which includes only the logarithms of the parton mass needed to regulate the collinear divergences [45], but are otherwise treated massless in the calculation, is not implemented.

DressedParticleSelector This selector takes a choice of dressing algorithm (cone or sequential recombination) and (flavourdependent) dressing parameters (cone radius or radial parameter and exponent). All charged particles of the process are then dressed with all photons using the specified algorithm with the given (flavourdependent) parameters. Their four momenta are added such that four momentum is conserved. The dressed charged particles may no longer be onshell and the photons used to dress the charged particles are removed from the list of particles. The resulting list of particles and their momenta are then passed to all subselectors.

Jet_Selector This selector uses FastJet [61] to build jets from a given list of input particles. It takes a list of flavours that are considered as jet finding input particles, the jet finding algorithm and its parameters including phase space boundaries in \(p_\perp \), \(\eta \) or y, as well as a minimal and maximal number of jets to be found as arguments. Additionally, clustered jets can be tagged or antitagged based on their flavour content, including relative and absolute constituent momentum requirements (e.g. b tagging a jet if one of its constituents is a bquark or anti\(\gamma \)tagging a jet if its photon constitents carry in excess of \(z_\text {thr}\) fraction of the total jet momentum). The clustered jets as well as all particles not used as jet finding input are passed on to all subselectors.

Isolation_Selector This selector uses the smooth cone isolation of [62] to isolate particles of a given flavour against particles of another flavour. It takes both the isolation flavour and rejection flavour list as well as the algorithm parameters as input. It further can be specified how many isolated particles of the given flavour should minimally and maximally be found in a given phase space volume (bounded by \(p_\perp \) and \(\eta \) or y ranges), e.g. exactly two isolated photons with \(p_\perp >30\,\text {GeV}\) and \(\eta <2.35\). The list of found isolated flavours and all other particles except for those that should be isolated, but are not, are passed to all subselectors.
Identified particles should in principle be defined using fragmentation functions, denoted \(D_i^j(z)\) for finding parton i in parton j at momentum fraction z. As, however, all \(D_i^i(z)\) have a \(\delta (1z)\) as leading term in onshell renormalisation schemes (and only differ by ratios of couplings in other schemes) simplified schemes exist that are applicable to many practical situations. Hence, no fragmentation function is implemented yet.
3.5 Flavour scheme conversion
4 Checks of the implementation
The implementation of the formalism described in the previous sections in Sherpa needs to be validated. To this end, both its independence of its internal free parameters, \(\{\alpha _\text {dip}\}=\{\alpha _\text { FF},\alpha _\text { FI},\alpha _\text { IF},\alpha _\text { II}\}\), \(\kappa \) and the choice of spectator in photon splittings, and its agreement with independent implementations for fixed values of these parameters need to be tested. While the latter were carried out in [19, 21, 25, 31, 32] against the private implementations in Munich [74], in [51] against MadGraph5 [75], and in [51, 53] against Recola [76] and found full agreement, the former represent a powerful check of internal consistency. Further, as the \(\{\alpha _\text {dip}\}\) parameters regulate the phase space coverage of the differential subtraction terms, they can be used to lower the average number of contributing dipoles for a given real emission contribution and, thus, reduce the computational costs of the realsubtracted contribution.
 (a)
The \({{\varvec{I}}}\) operator of Sect. 2.2 containing the explicit Laurent expansion in \(\epsilon \) as \(\epsilon \rightarrow 0\) needs to reproduce the correct \(\epsilon ^{2}\) and \(\epsilon ^{1}\) coefficients in order to cancel all corresponding poles of the virtual matrix elements, leading to a finite integrand in Eq. (2.2) in \(d=4\). These checks were performed for all possible dipoles in [19, 21, 25, 31, 32, 34, 37, 51, 52, 53] and are not repeated here.
 (b)The expectation value of any infrared observable is independent of the choice of the technical \(\{\alpha _\text {dip}\}\) and \(\kappa \) parameters as well as the choice \(c_{\tilde{k}}^\gamma \) of spectator for photon splittings. It thus needs to be verified that the sum of the contributions of the \(\epsilon ^0\) coefficient of the integrated subtraction terms and the differential subtraction terms is independent of these parameters and choices. In order to arrive at a finite result in \(d=4\), cf. Eq. (2.2), the real emission correction and the collinear counterterms are added. The corresponding quantity is defined asand will be evaluated for processes containing all available dipole configurations in the following.$$\begin{aligned} \langle O\rangle _\text {IRD}&= \int \mathrm{d}\Phi _m^{(4)}\; \left[ \mathrm {I}_\mathrm {D}(\Phi _m^{(4)};\{\alpha _\text {dip}\},\kappa ,c_{\tilde{k}}^\gamma )\right. \nonumber \\&\qquad \left. +\,\mathrm {C}(\Phi _m^{(4)}) \right] _{\epsilon ^0\,\text {coeff.}} O(\Phi _m^{(4)})\nonumber \\&\quad +\int \mathrm{d}\Phi _{m+1}^{(4)} \left[ \mathrm {R}(\Phi _{m+1}^{(4)})\;O(\Phi _{m+1}^{(4)})\right. \nonumber \\&\qquad \left. \,\mathrm {D}(\Phi _m^{(4)}\cdot \Phi _1^{(4)};\{\alpha _\text {dip}\},\kappa ,c_{\tilde{k}}^\gamma )\;O(\Phi _m^{(4)}) \right] , \end{aligned}$$(4.1)
Numerical values of all input parameters. The gauge boson masses are taken from [72], while their widths are obtained from stateof the art calculations. The Higgs mass and width are taken from [73]. The top quark mass is taken from [72] while its width has been calculated at NLO QCD. In calculations where a massive particle is present as an external state, its width is set to zero
\(G_\mu =1.1663787\times 10^{5}~\text {GeV}^2\)  
\(m_W=80.385~\text {GeV}\)  \(\Gamma _W=2.0897~\text {GeV}\) 
\(m_Z=91.1876~\text {GeV}\)  \(\Gamma _Z=2.4955~\text {GeV}\) 
\(m_t=173.2~\text {GeV}\)  \(\Gamma _t=1.339~\text {GeV}\) 
Massless dipoles. Contrary to the QCD case, in the Standard Model almost all particles carry QED charges and therefore participate in the construction of dipoles. One notable exception are neutrinos. Therefore, in order to investigate the behaviour of the massless II, IF, FI and FF dipoles, and, thus, the independence of \(\alpha _\text { II}, \alpha _\text { IF}, \alpha _\text { FI}\) and \(\alpha _\text { FF}\) separately in this technical validation \(\sigma _\text {IRD}\) is considered for all three different rotations of the interaction of a quarkantiquark pair with a neutrinoantineutrino pair. Hence, besides the \(\sigma _\text {IRD}\) contribution to the \(\mathcal {O}(\alpha )\) correction to \(pp\rightarrow \nu _e\bar{\nu }_e\) at the LHC at an invariant mass of 13 TeV, \(\sigma _\text {IRD}\) is computed for both the production of at least two jets at a hypothetical \(\nu _e\bar{\nu }_e\) collider at a centreofmass energy of 1 TeV and inclusive single jet production in equally hypothetical \(\nu _ep\) deep inelastic scattering (DIS) with the same centreofmass energy are calculated.
Moving away from the simplest configurations, Fig. 3 displays the results for electronpositron pair production. The left hand side plot again displays their production at the hypothetical \(\nu _e\)\(\bar{\nu }_e\) collider used before, finding very similar results and their independence of \(\alpha _\text { FF}\). The centre plot now, however, displays the production of an electronpositron pair at the LHC. At leading order, this process proceeds through \(q\bar{q}\rightarrow e^e^+\) and \(\gamma \gamma \rightarrow e^e^+\) at \(\mathcal {O}(\alpha ^2)\). Consequently, the \(q\bar{q}\) channel exhibits six dipoles of all four types. In the \(\gamma \gamma \) channel, the number and types of dipoles present depends on the choice of possible photon splitting spectators \(c_{\tilde{k}}^\gamma \). To regulate all LO singularities the fiducial phase space is defined by requiring the dressed electrons to have a transverse momentum of at least \(20\,\text {GeV}\) and the pair to have an invariant mass of at least \(60\,\text {GeV}\). As \(\sigma _\text {IRD}\) now potentially depends on the full set \(\{\alpha _\text {dip}\}\) no continuous variation over four orders of magnitude is performed. Instead, each of the four parameters is varied independently to 0.001 keeping all others at their default value of 1. These four variations are completed by setting \(\alpha _\text { FF}=\alpha _\text { FI}=\alpha _\text { IF}=\alpha _\text { II}=1\) and 0.001. The resulting correction contributions are found to be independent of \(\{\alpha _\text {dip}\}\).
The left plot of Fig. 5 now investigates the dependence of \(\sigma _\text {IRD}\) on the \(\kappa \) parameter. In only arises in FF dipoles of gluons splitting into gluons or massless quarks or photons splitting into massless fermions in the presence of a massive spectator. Consequently, to restrict the number of additional contributions, topantitop pair production in association with one jet at the hypothetical \(\nu _e\)–\(\bar{\nu }_e\) collider is considered. At LO, this process contributes is defined both at \(\mathcal {O}(\alpha _s\alpha ^2)\) and \(\mathcal {O}(\alpha ^3)\) where the final state photon forms the jet. At NLO, there are contributions at \(\mathcal {O}(\alpha _s^2\alpha ^2), \mathcal {O}(\alpha _s\alpha ^3)\) and \(\mathcal {O}(\alpha ^4)\). The \(\mathcal {O}(\alpha _s\alpha ^3)\) contribution, however, contains neither gluon nor photon splittings. Due to the relative size of \(g\rightarrow gg\) and \(g\rightarrow q\bar{q}\) splittings in relation to gluon radiation off the top quarks the \(\kappa \) dependence of the \(\mathcal {O}(\alpha _s^2\alpha ^2)\) contribution is much more pronounced than at \(\mathcal {O}(\alpha ^4)\), where photon radiation off the top quarks overwhelms the photon splitting contribution. Nonetheless, at both orders an independence of the \(\sigma _\text {IRD}\) of \(\kappa \) is found. In addition, the right plot investigates the influence on the choice of spectators for the final state photon splitting ocurring at \(\mathcal {O}(\alpha ^4)\). No dependence on this choice is observed. Please note that for this process scheme 0 and 3 as well as scheme 1 and 2 lead to the same set of allowed spectators, respectively, and therefore to identical results.
External \(\varvec{W}\) bosons. Lastly, it may become necessary to also consider external W bosons (or other massive charged particle with spin \(>\tfrac{1}{2}\) in BSM theories) as stable final state particles, e.g. to reduce the computational complexity for high final state multiplicity processes where offshell effects and effects in the decays can be ignored or recovered through other means [83]. In this case the literature does not provide expressions for the respective massive dipole functions. As their mass, however, can be assumed to be large enough to suppress collinear radiation well enough, this only leaves the spinindependent soft photon emission limit.^{3} Here, both the expressions for the radiation of a photon off a massive scalar or a massive fermion can be used.
Figure 7 details the production of a \(W^+W^\) pair in the hypothetical \(\nu _e\bar{\nu }_e\) collider, separating the FF dipoles and their \(\alpha _\text { FF}\) dependence, on the left hand side. As before, \(\sigma _\text {IRD}\) is found to be independent within the statistical accuracy and also independent of the whether the massive scalar or massive fermion subtraction terms are used. The right hand side focusses on their production at the LHC. Again, all dipoles contribute at \(\mathcal {O}(\alpha ^3)\), leading again to the afore described sixpoint variation. Also in this case, the result is independent of \(\{\alpha _\text {dip}\}\) and the choice of scalar or fermionic subtraction term. The FI and IF dipoles cannot be investigated separately, as in the \(\nu _ep\rightarrow \nu _et\) case, due to charge conservation.
5 Conclusions
This paper detailed the construction and implementation of the adaptation of the Catani–Seymour subtraction formalism for NLO EW calculations. Besides the translation of the QCD dipole functions to the QED case, several other issues have been addressed. They include the special role photon splittings play in the formalism, embedding extermal massive emitters of spin \(>\tfrac{1}{2}\) into the formalism and the interplay of QCD and QED subtractions for processes exhibiting both kinds of divergences. The resulting general subtraction for NLO EW calculations has been implemented in the Sherpa MonteCarlo event generator framework. Interfaces to OpenLoops, GoSam and Recola to access the needed virtual corrections exist and are fully functional.
In addition to the checks against independent implementations on the level of partial and total cross sections performed in previous publications, numerous internal cross checks for independence of technical parameter choices, \(\{\alpha _\text {dip}\}=\{\alpha _\text { FF},\alpha _\text { FI},\alpha _\text { IF},\alpha _\text { II}\}\), \(\kappa \) and the choice of spectator in photon splittings \(c_{\tilde{k}}^\gamma \), have been presented here. This implementation will become publically available in the near future with the next major Sherpa release and an extension to the Comix matrix element generator [84] is foreseen.
Footnotes
Notes
Acknowledgements
MS would like to thank N. Greiner, S. Höche, S. Kallweit, J. Lindert, and S. Schumann for numereous checks of the implementation and S. Pozzorini for many useful discussion. MS acknowledges support by the Swiss National Foundation (SNF) under contract PP00P212855 as well as the European Union’s Horizon 2020 research and innovation programme as part of the Marie SkodowskaCurie Innovative Training Network MCnetITN3 (Grant agreement no. 722104) and the ERC Advanced Grant MC@NNLO (340983). MS would also like to thank the University DuisburgEssen for the kind hospitality during essential parts of the project. All plots are based on Rivet ’s [85] plotting tools.
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